We prove that for every $t\in \mathbb{N}$ there is a constant $γ_t$ such that every graph with twin-width at most $t$ and clique number $ω$ has chromatic number bounded by $2^{γ_t \log^{4t+3} ω}$. In other words, we prove that graph classes of bounded twin-width are quasi-polynomially $χ$-bounded. This provides a significant step towards resolving the question of Bonnet et al. [ICALP 2021] about whether they are polynomially $χ$-bounded.